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G. E. KISILEVSKII
CONDITIONS FOR THE NONDECOMPOSABILITY OF DISSIPATIVE VOLTERRA OPERATORS WITH FINITE-DIMENSIONAL IMAGINARY COMPONENT
(Presented by Academician L. S. Pontryagin on 2 VI 1964)
Let \(A\) be a simple Volterra dissipative\(^*\) operator acting in a separable Hilbert space \(\mathfrak H\). If
\[
\operatorname{sp}\frac{A-A^*}{2i}=l<\infty
\]
and \(W(z)\) is the characteristic matrix-function (c.m.f.) of the operator \(A\), then \((^2)\)
\[
\|W(z)\|\le e^{2l|z|}. \tag{1}
\]
M. S. Brodskii showed \((^2)\) that if equality is attained in estimate (1), then the operator \(A\) is nondecomposable. In the present article some necessary conditions for nondecomposability are established.
- Let the imaginary component
\[ \frac{A-A^*}{2i} \]
of the operator \(A\) be \(n\)-dimensional \((2\le n<\infty)\). The c.m.f. of the operator \(A\) satisfies the following conditions:
1) \(W(z)\) is an entire matrix-function of the complex variable \(z\);
2) \(W(z)=I+izH+\ldots\), where \(H\) is a nonzero Hermitian nonnegative matrix-function;
3) \(W^*(z)W(z)=I\) \((\operatorname{Im} z=0)\);
4) \(W^*(z)W(z)\ge I\) \((\operatorname{Im} z<0)\).
A matrix-function \(W_1(z)\) is called a divisor of the c.m.f. \(W(z)\) if \(W(z)\) can be represented in the form
\[
W(z)=W_2(z)W_1(z),
\]
where \(W_k(z)\) \((k=1,2)\) satisfy conditions 1)—4). By virtue of \((^3)\), the operator \(A\) is nondecomposable if and only if its c.m.f. is ordered.
Lemma 1. If the characteristic matrix-function \(W(z)\) of the operator \(A\) has a scalar divisor of the form
\[
W_0(z)=e^{il_0z}I\quad (l_0>0),
\]
then the operator \(A\) is decomposable.
Proof. It suffices to show that the c.m.f. \(W(z)\) is not ordered, i.e., that there exist such divisors of it which are not divisors of each other. Such divisors, as is easy to see, are the matrix-functions
\[
W_k(z)=e^{il_0z}P_k\quad (k=1,2),
\]
where \(P_1\) and \(P_2\) are projection matrices of the same order as \(W(z)\), with
\[
P_1P_2=0.
\]
For entire matrix-functions, as also for scalar functions, one can introduce the concepts of order and type of growth.
Lemma 2. The order of growth of any characteristic matrix-function of the operator \(A\) is equal to \(1\), and the type of growth \(\sigma\) satisfies the inequality
\[
\frac{2l}{n}\le \sigma\le 2l.
\]
Proof. It is not hard to prove that all c.m.f.’s of the operator \(A\) have the same growth. Therefore, without loss of generality, one may
\[ \text{\(^*\) The terminology introduced in \((^1)\) is used in the article.} \]
consider that the order of the c.m.f. \(W(z)\) is equal to \(n\). Denoting by \(\lambda_1^2 \leqslant \lambda_2^2 \leqslant \cdots \leqslant \lambda_n^2\) the eigenvalues of the matrix \(W^*(z)W(z)\) and taking into account that \(\det W(z)=e^{2ilz}(1)\), we obtain
\[ \|W(z)\|^2=\|W^*(z)W(z)\|= \]
\[ =\lambda_n^2 \geqslant \sqrt[n]{\lambda_1^2\lambda_2^2\cdots\lambda_n^2} =\sqrt[n]{\det[W^*(z)W(z)]}=e^{-\frac{4l}{n}\operatorname{Im}z}. \]
Thus, for all \(z\) the inequality
\[ \|W(z)\|\geqslant e^{-\frac{2l}{n}\operatorname{Im}z} \tag{2} \]
holds.
The assertion of the lemma follows from inequalities (1) and (2).
Theorem 1. If the growth type \(\sigma\) of the c.m.f. \(W(z)\) of the operator \(A\) satisfies the inequality
\[ \frac{2l}{n}\leqslant \sigma<\frac{2l}{n-1}, \]
then the operator \(A\) is not unicellular.
Proof. We shall assume that the order of the c.m.f. \(W(z)\) is equal to \(n\). Since \(\|W(z)\|=1\) \((\operatorname{Im}z=0)\), by generalizing to matrix functions a well-known theorem from the theory of entire scalar functions \(([4], \text{Chapter I, Theorem 22})\), we obtain the inequality
\[ \|W(z)\|\leqslant e^{-\sigma\operatorname{Im}z}\qquad (\operatorname{Im}z\leqslant 0). \tag{3} \]
Let \(\lambda_1^2\leqslant\lambda_2^2\leqslant\cdots\leqslant\lambda_n^2\) be the eigenvalues of the matrix \(W^*(z)W(z)\). From the equality \(\lambda_1^2\lambda_2^2\cdots\lambda_n^2=e^{-4l\operatorname{Im}z}\) it follows that
\[ \lambda_1^2=\frac{1}{\lambda_2^2\cdots\lambda_n^2}\,e^{-4l\operatorname{Im}z} \geqslant \frac{1}{\lambda_n^{2(n-1)}}\,e^{-4l\operatorname{Im}z}. \]
From (3) it follows that
\[ \lambda_n^2=\|W^*(z)W(z)\|=\|W(z)\|^2 \leqslant e^{-2\sigma\operatorname{Im}z}\qquad (\operatorname{Im}z\leqslant 0) \]
and, consequently,
\[ \lambda_1^2\geqslant e^{-2[2l-\sigma(n-1)]\operatorname{Im}z} \qquad (\operatorname{Im}z\leqslant 0). \]
Introducing the notation \(l_0=2l-\sigma(n-1)\) and taking into account that \(\lambda_1^2\) is the smallest eigenvalue of the matrix \(W^*(z)W(z)\), we obtain the inequality
\[ W^*(z)W(z)\geqslant e^{-2l_0\operatorname{Im}z}I \qquad (\operatorname{Im}z\leqslant 0). \tag{4} \]
Since \(\sigma<\dfrac{2l}{n-1}\), we have \(l_0>0\), and the scalar matrix function \(W_0(z)=e^{il_0z}I\) satisfies conditions 1)—4). We shall show that the matrix function \(W_1(z)=W(z)W_0^{-1}(z)\) also satisfies these conditions. The fulfillment of conditions 1)—3) is obvious. To verify condition 4), consider the equality
\[ W_1^*(z)W_1(z)=W_0^{*-1}(z)\,W^*(z)W(z)\,W_0^{-1}(z)= \]
\[ =e^{il_0\bar z I}\,W^*(z)W(z)\,e^{-il_0zI} =e^{2l_0\operatorname{Im}z}W^*(z)W(z), \]
whence, by virtue of (4), the inequality
\[ W_1^*(z)W_1(z)\geqslant I\qquad (\operatorname{Im}z<0) \]
follows.
Thus, \(W(z)\) has the scalar divisor \(W_0(z)\), and, consequently, the operator \(A\) is not unicellular.
Consider the case when \(\sigma=2l/n\). From (2) and (3) it follows that
\[ \|W(z)\|=e^{-\frac{2l}{n}\operatorname{Im}z} \qquad (\operatorname{Im}z\leqslant 0). \]
Therefore, as is easy to prove,
\[ \lambda_1^2=\lambda_2^2=\ldots=\lambda_n^2=e^{-\frac{4l}{n}\operatorname{Im} z} \qquad (\operatorname{Im} z \leq 0), \]
i.e., the matrix-function \(W^*(z)W(z)\) is scalar in the lower half-plane. Since \(W(z)=W^{*-1}(\bar z)\) \((^1)\), it follows that
\[ \begin{aligned} \|W(z)\|^2 &=\|W^*(z)W(z)\| =\|W^{-1}(\bar z)W^{*-1}(\bar z)\| \\ &=\|[W^*(\bar z)W(\bar z)]^{-1}\| =e^{\frac{4l}{n}\operatorname{Im}\bar z} =e^{-\frac{4l}{n}\operatorname{Im}z} \qquad (\operatorname{Im} z>0). \end{aligned} \]
Thus, the equality
\[ \|W(z)\|=e^{-\frac{2l}{n}\operatorname{Im}z} \]
holds for all \(z\). Considering the scalar matrix-function
\[ W_0(z)=e^{\frac{2ilz}{n}} I, \]
we obtain that \(\|W(z)W_0^{-1}(z)\|\equiv 1\), whence it follows that the matrix-function \(W(z)W_0^{-1}(z)\) is identically equal to some constant matrix. From condition 2 it is clear that \(W(z)W_0^{-1}(z)\equiv I\); consequently,
\[ W(z)=e^{\frac{2ilz}{n}} I . \tag{5} \]
Considering the triangular model \((^1)\) of the operator with characteristic matrix-function (5), we obtain the following result.
Theorem 2. If \(\sigma=2l/n\), then the space \(\mathfrak H\) decomposes into the orthogonal sum \(\mathfrak H=\mathfrak H_1\oplus\ldots\oplus\mathfrak H_n\) of subspaces invariant with respect to the operator \(A\), in each of which the induced operator is unicellular and has a one-dimensional imaginary component.
Consider the multiplicative representation \((^1)\) of the characteristic matrix-function
\[ W(z)=\int_0^l e^{2izt}\,dH(t) \left( H(t)=\int_0^t B(x)\,dx,\quad B^*(x)=B(x),\quad B(x)\geq 0,\quad \operatorname{sp}B(x)\equiv 1 \right). \tag{6} \]
The idea of the proof of the following assertion belongs to Yu. L. Shmul’yan.
Lemma 3. If the rank of the matrix-function \(B(t)\) is equal to \(n\) on a set of positive measure, then the operator \(A\) is not unicellular.
Proof. Let \(m(t)\) be the lower bound of the spectrum of the matrix \(B(t)\); then \(m(t)>0\) \((t\in e_0\subset[0,l],\ \operatorname{mes} e_0>0)\) and
\[ l_0=\int_0^l m(t)\,dt>0. \]
It is not difficult to prove that the scalar matrix-function
\[ W_0(z)=e^{2il_0z}I \]
is a divisor of the characteristic matrix-function \(W(z)\), and, consequently, the operator \(A\) is not unicellular.
- Consider the case \(n=2\). Comparing the criterion of M. S. Brodskii with Lemma 2 and Theorem 1, we obtain the following results.
Theorem 3. In order that a simple Volterra dissipative operator with a two-dimensional imaginary component be unicellular, it is necessary and sufficient that the type of growth \(\sigma\) of its characteristic matrix-function be equal to the non-Hermitian trace \(2l\) of the operator.
Theorem 4. In order that a simple Volterra dissipative operator with a two-dimensional imaginary component be unicellular, it is necessary and sufficient that its characteristic matrix-function of the second order have no scalar divisors.
Theorem 5. For any simple Volterra dissipative operator with a two-dimensional imaginary component there exists a spectral function \((^5)\) of rank one.
Proof. It suffices to prove that the c. m.-f. of the operator is representable in the form (6), where the rank of the matrix-function \(B(t)\) is equal to 1 almost everywhere. In the case \(\sigma=2l\) this follows from Lemma 3, since, by M. S. Brodskii’s criterion, the operator is unicellular.
If \(\sigma=l\), then
\[ W(z)=e^{ilzI}=e^{ilzP_2}e^{ilzP_1}=\int_0^l e^{2iz}\,dH(t), \]
where
\[ P_1+P_2=I,\qquad P_1P_2=0,\qquad H(t)=\int_0^t B(x)\,dx, \]
\[ B(x)= \begin{cases} P_1, & \text{for } 0\le t<l/2,\\ P_2, & \text{for } l/2\le t\le l. \end{cases} \]
Let \(l<\sigma<2l\); then \(W(z)=e^{il_0 zI}W_1(z)\) \((l_0=2l-\sigma)\). It can be shown that \(W_1(z)\) has no scalar divisors and, consequently, is regular; therefore, in the multiplicative representation
\[ W_1(z)=\int_0^l e^{2iz}\,dH_1(t) \left( H_1(t)=\int_0^t B_1(x)\,dx,\quad l_1=\sigma-l \right) \]
the rank of the matrix-function \(B_1(t)\) is equal to 1 almost everywhere. Defining the matrix-function \(B(t)\) by the equalities
\[ B(t)= \begin{cases} B_1(t), & \text{for } 0\le t<l_1,\\ P_1, & \text{for } l_1\le t<l_1+\tfrac12 l_0,\\ P_2, & \text{for } l_1+\tfrac12 l_0\le t\le l_1+l_0, \end{cases} \qquad (P_1+P_2=I,\quad P_1P_2=0) \]
we obtain the obvious representation
\[ W(z)=\int_0^l e^{2iz}\,dH(t) \left( H(t)=\int_0^t B(x)\,dx,\quad l=l_1+l_0 \right). \]
Let \(A\) be a simple Volterra dissipative operator with two-dimensional imaginary component, and let \(W(z)\) be its c. m.-f. of the second order. Then 3 cases are possible:
I. \(\sigma=2l\); in this case the operator \(A\) is unicellular, and the c. m.-f. \(W(z)\) is regular.
II. \(\sigma=l\); in this case the operator decomposes into an orthogonal sum of two unicellular operators, and the c. m.-f. \(W(z)\) is scalar.
III. \(l<\sigma<2l\); in this case the operator \(A\) has two mutually orthogonal invariant subspaces, and the c. m.-f. is representable in the form \(W(z)=W_0(z)W_1(z)\), where \(W_1(z)\) is regular and \(W_0(z)\) is scalar.
Zhytomyr Pedagogical Institute
named after I. Franko
Received
28 V 1964
CITED LITERATURE
\(^{1}\) M. S. Brodskii, M. S. Livshits, UMN, 13, no. 1 (79), 3 (1958).
\(^{2}\) M. S. Brodskii, DAN, 111, No. 5, 926 (1956).
\(^{3}\) M. S. Brodskii, DAN, 138, No. 3, 512 (1961).
\(^{4}\) B. Ya. Levin, Distribution of Zeros of Entire Functions, 1956, p. 70.
\(^{5}\) M. S. Brodskii, UMN, 16, no. 1 (97), 135 (1961).